Let $X$ be the set of positive natural numbers. Define the following topology $\tau$ on $X$: $$A\in \tau \iff 1\notin A \,\, or \,\, 1\in A \,\, and \,\,\lim\limits_{n\to\infty}\dfrac{S(n,A)}{n}=1$$.
Where $S(n,A)$ is the number of elements in $A$ which are less then or equal to $n$. I was wondering how do I know if, given $A\subset X$, this limit actually exists?
It depends on how $A$ is given ( e g a formula) and whether $S(A,n)$ is computable for all or sufficiently many $n$. But the limit exists or not, and if it exists it’s $1$ or less. So the topology itself is well-defined.
E.g. for any bounded sequence of reals, like $a_n=\frac{S(A,n)}{n}$, for which we know that $0 \le a_n \le 1$ for all $n$, $\liminf_n a_n$ and $\limsup_n a_n$ exist in $[0,1]$ (recall your analysis class); if they're botht equal to $1$ then $\lim_{n \to \infty} a_n = 1$ and $A \cup \{1\}$ is a neighbourhood of $1$, otherwise not.
It's clearly well-defined for easy sets like $\Bbb N$ or $\Bbb N\setminus F$, where $F$ is finite (in both cases indeed $1$) and the even integers (value $\frac12$). It's actually a well studied notion of the density of a set of integers taht's being used here.